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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Solve the differential equation.

8. d 2 y d x 2 + 4 y = sin 2 x

To determine

To solve: The differential equation.

Explanation

Given data:

The differential equation is,

d2ydx2+4y=sin2x

y+4y=sin2x (1)

Consider the auxiliary equation.

r2+4=0 (2)

Roots of equation (2) are,

r=0±(0)24(1)(4)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=±i42=±2i

Write the expression for the complementary solution of two complex roots r=α±iβ .

yc(x)=eαx(c1cosβx+c2sinβx) (3)

Substitute 0 for α and 2 for β in equation (3),

yc(x)=e0x(c1cos2x+c2sin2x)

yc(x)=c1cos2x+c2sin2x (4)

The Right hand side (RHS) of a differential equation contains only sine function, Therefore, the trail solution yp(x) for this case can be expressed as follows.

yp(x)=Axcos2x+Bxsin2x (5)

Differentiate equation (5) with respect to x.

yp(x)=ddx(Axcos2x+Bxsin2x)

yp(x)=(A+2Bx)cos2x+(B2Ax)sin2x (6)

Differentiate equation (6) with respect to x

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